Saudi Arabia cut off diplomatic relations with Iran on Sunday, giving Iranian diplomats 48 hours to leave the country, as the kingdom’s execution of a dissident cleric inflamed sectarian tensions in a region already brimming with conflict.

Crowds of incensed Shiite Muslims demonstrated in cities from Tehran to Beirut, condemning the death of Nemer al-Nemer—a prominent Shiite cleric put to death in the mass execution of 47 prisoners on Saturday—and calling for the downfall of the Saudi government. Iranian Supreme Leader...